Both integrals can be solved by substitution, and while I am comfortable with that, in both cases I find the method unbearably ugly, mostly because there are hundreds of overtly feasible substitutions (and the corresponding factor the denominator and numerator is multiplied by) a when you look at the integral for the very first time, and so the one that happens to work must be memorised, either by rote or experience using it.

Is there a faster or more aesthetically appealing method of computing these (types of) integrals that 'forces the answer upon you' to a greater extent so that the solution does not require bursts of insight or previous experience, and can be applied generally to many types of awkward trigonometric integrals? Something using complex analysis maybe?

That's the sort of thing I'm trying to avoid, as you need a unique epiphany for each integrand.
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AlyoshaJan 1 '13 at 17:11

1

definitely you would need experience of patters to calculate integrals. The easiest way ... memorize the results. differentiation of the result will always give you a way which you will never miss.
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Santosh LinkhaJan 1 '13 at 17:13

I think the simpliest way is$$\int\frac{dx}{\cos x}=\int\frac{\cos x dx}{\cos^2 x}=\int\frac{d(\sin x)}{1-\sin^2 x}=\int\frac{dz}{1-z^2}=\frac{1}{2}\left(\int\frac{dz}{1-z}+\int\frac{dz}{1+z}\right)$$ Now these are easy to calculate. The same method works for the other integral also. Generally if you have a integral like $$\int\frac{dx}{\cos x\cdot F(\sin x)}$$ where $F$ is some nice polynomial ( like product of linear and quadratic factors), then you can use the same trick to convert the integral into the form$$\int\frac{dz}{(1-z^2)\cdot F(z)}$$ and try to solve it using the method of partial fractions.

Let we have $\int R(\sin(x),\cos(x))~dx$ wherein $R$ is a rational function respect to $\sin(x), \cos(x)$. Then we have the following substations also: $$R(-\sin(x),\cos(x))\equiv -R(\sin(x),\cos(x))\Longrightarrow t=\cos(x)\\\ R(\sin(x),-\cos(x))\equiv -R(\sin(x),\cos(x))\Longrightarrow t=\sin(x)$$